Applying Chaos Control in Periodic Windows

Methods developed for the control of chaos usually consider a nonlinear system set at particular parameters which result in chaotic motion. Then, using only tiny control adjustments, a chaotic trajectory is stabilised onto a choice of unstable periodic solutions embedded within the original chaotic motion. In this paper however we make use of the periodic windows, which typically exist within the chaotic regimes of a bifurcation diagram as a single parameter is varied, so that the same advantages of chaos control can be achieved, but now with the system initially set at parameters which produce a stable periodic motion. Within a periodic window an infinite number of unstable solutions may naturally coexist with the stable state. By perturbing the stable motion a chaotic-like transient can be induced which, when it approaches the location of a desired solution, can be stabilised. We illustrate the feasibility of this idea using a self-locating control scheme in a periodic window of the Hénon map. We show that, starting from a stable periodic motion, one can flexibly manipulate the system between many co-existing solutions without major changes to the overall configuration of the system